Positive integer solutions to $(x+y)^2+(y+z)^2+(z+x)^2=12xyz$

TL;DR

This paper characterizes positive integer solutions to a specific Diophantine equation and introduces a generalized cluster pattern to describe these solutions.

Contribution

It provides a novel description method for solutions and uncovers a generalized cluster pattern underlying the equation.

Findings

Explicit description of positive integer solutions

Introduction of a generalized cluster pattern

New insights into the structure of solutions

Abstract

In this paper, we give a specific way of describing positive integer solutions of a Diophantine equation $(x+y)^2+(y+z)^2+(z+x)^2=12xyz$ and introduce a generalized cluster pattern behind it.